| L(s) = 1 | − 2·3-s + 9-s + 6·11-s − 6·17-s − 5·25-s + 4·27-s − 12·33-s − 6·41-s − 10·43-s − 7·49-s + 12·51-s − 6·59-s + 14·67-s − 2·73-s + 10·75-s − 11·81-s + 18·83-s + 18·89-s − 10·97-s + 6·99-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s + 1.80·11-s − 1.45·17-s − 25-s + 0.769·27-s − 2.08·33-s − 0.937·41-s − 1.52·43-s − 49-s + 1.68·51-s − 0.781·59-s + 1.71·67-s − 0.234·73-s + 1.15·75-s − 1.22·81-s + 1.97·83-s + 1.90·89-s − 1.01·97-s + 0.603·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.05898886788983, −13.42160723649422, −13.25949991090262, −12.28459768866054, −12.06809826405055, −11.65331187718642, −11.15772025818490, −10.88206300228725, −10.18345505696006, −9.541847403280052, −9.274498014575438, −8.530400217747662, −8.215241540059248, −7.322490368603182, −6.648056063877048, −6.509021622787336, −6.095905348223645, −5.284653787862192, −4.866875920563718, −4.252063513267925, −3.737056096274089, −3.100396946115008, −2.044895114257728, −1.626656807127209, −0.7401721901715944, 0,
0.7401721901715944, 1.626656807127209, 2.044895114257728, 3.100396946115008, 3.737056096274089, 4.252063513267925, 4.866875920563718, 5.284653787862192, 6.095905348223645, 6.509021622787336, 6.648056063877048, 7.322490368603182, 8.215241540059248, 8.530400217747662, 9.274498014575438, 9.541847403280052, 10.18345505696006, 10.88206300228725, 11.15772025818490, 11.65331187718642, 12.06809826405055, 12.28459768866054, 13.25949991090262, 13.42160723649422, 14.05898886788983
